What is the difference between asymmetric and depthwise separable convolution?
They are not the same thing.  

asymmetric convolutions work by taking the x and y axes of the image separately. For example performing a convolution with an $(n \times 1)$ kernel before one with a $(1 \times n)$ kernel.  

On the other-hand depth-wise separable convolutions separate the spatial and channel components of a 2D convolution. It will first perform the $(n \times n)$ convolution on each channel separately (full kernel shape will be $(n \times n \times 1)$ rather than $(n \times n \times k)$ where $k$ is the number of channels in the previous layer) before doing a $(1 \times 1)$ convolution to learn a relationship between the channels (full kernel size for that being $(1 \times 1 \times k)$)